On Area Comparison and Rigidity Involving the Scalar Curvature

We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting result follows from an area comparison theorem for hypersurfaces with non-positive Sigma-constant (Theorem 4) that generalises [23, Theorem 2]. Finally, we will address the optimality of these comparison and splitting results by explicitly constructing several examples

On area comparison and rigidity involving the scalar curvature

In this thesis we study the effects of lower bounds for the curvature of a Riemannian manifold M on the geometry and topology of closed, minimal hypersurfaces. We will prove an area comparison theorem for totally geodesic surfaces which is an optimal analogue of the Heintze-Karcher-Maeada Theorem in the context of 3-manifolds with lower bounds on scalar curvature (Theorem 3.8). The optimality of this result will be addressed by explicitly constructing several counterexamples in dimensions n ≥ 4. This area comparison theorem turns out that it provides a unified proof of three splitting and rigidity theorems for 3-manifolds with lower bounds on the scalar curvature that were first proved, independently, by Cai-Galloway, Bray-Brendle- Neves and Nunes (Theorem 4.7 (a)-(c)). In the final part of this thesis we will address some natural higher dimensional generalisations of these splitting and rigidity results and emphasise some connections with the Yamabe problem

Technological challenges to understanding the microbial ecology of deep subsurface ecosystems

Peer Reviewe